# Inverse of strictly diagonally dominant matrix

I have a matrix whose diagonal entries are positive whereas non-diagonal entries are negative.This matrix is also Strictly diagonally dominant.

Can we say that all elements of the inverse of this matrix is strictly positive i.e $$a_{ij}$$>0 .

Yes. Scale $$A$$ by a positive factor and we may assume that $$\max_ia_{ii}<1$$. Then $$B:=I-A$$ is positive and $$\sum_j|b_{ij}| =|b_{ii}|+\sum_{j\ne i}|b_{ij}| =1-a_{ii}+\sum_{j\ne i}|a_{ij}| <1$$ for each $$i$$. Hence $$\|B\|_\infty<1$$ and we may expand $$A^{-1}=(I-B)^{-1}$$ as an infinite sum $$I+B+B^2+\ldots$$. However, as $$B$$ is positive, the infinite sum is positive too.